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Consider the projection operators $P_1, P_2, P_1', P_2'$ with $P_1+P_2=I$, and $\lambda,\mu\in\mathbb C$ with $\lambda\neq\mu$.

Does $\lambda P_1+\mu P_2=\lambda P_1'+\mu P_2'$ imply $P_1'=P_1$ and $P_2'=P_2$?

Does the result change if we instead use real coefficients $\lambda,\mu\in\mathbb R$? Do we need the projectors to be self-adjoint for the result to hold?

A similar problem with projection operators was addressed in this question, but in those cases they didn't have additional constants multiplying the projections.

glS
  • 7,963

1 Answers1

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If you have that $P_1' + P_2' = I$ as well, then we can make the following simplification:

$$(\lambda - \mu)P_1 + \mu P_1 + \mu P_2 = (\lambda - \mu) P_1 + \mu I = (\lambda - \mu)P_1' + \mu I$$

by making the same simplification on the right hand side. Therefore since $\lambda \neq \mu$, we have

$$(\lambda - \mu) P_1 = (\lambda - \mu) P_1' \implies P_1 = P_1'$$

Ninad Munshi
  • 37,891
  • do we have to assume that though? I feel like the result should hold more generally – glS Sep 17 '19 at 00:00